↑

Properties

Label 2-160-20.7-c5-0-7
Degree $2$
Conductor $160$
Sign $0.881 - 0.472i$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−17.6 − 17.6i)3-s + (55.7 + 3.44i)5-s + (52.6 − 52.6i)7-s + 381. i·9-s + 126. i·11-s + (−788. + 788. i)13-s + (−924. − 1.04e3i)15-s + (425. + 425. i)17-s − 182.·19-s − 1.86e3·21-s + (−846. − 846. i)23-s + (3.10e3 + 384. i)25-s + (2.44e3 − 2.44e3i)27-s + 6.17e3i·29-s + 7.36e3i·31-s + ⋯
L(s)  = 1  + (−1.13 − 1.13i)3-s + (0.998 + 0.0616i)5-s + (0.406 − 0.406i)7-s + 1.56i·9-s + 0.315i·11-s + (−1.29 + 1.29i)13-s + (−1.06 − 1.20i)15-s + (0.357 + 0.357i)17-s − 0.116·19-s − 0.920·21-s + (−0.333 − 0.333i)23-s + (0.992 + 0.123i)25-s + (0.644 − 0.644i)27-s + 1.36i·29-s + 1.37i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.881 - 0.472i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.881 - 0.472i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.191295764\)
\(L(\frac12)\) \(\approx\) \(1.191295764\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-55.7 - 3.44i)T \)
good3 \( 1 + (17.6 + 17.6i)T + 243iT^{2} \)
7 \( 1 + (-52.6 + 52.6i)T - 1.68e4iT^{2} \)
11 \( 1 - 126. iT - 1.61e5T^{2} \)
13 \( 1 + (788. - 788. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-425. - 425. i)T + 1.41e6iT^{2} \)
19 \( 1 + 182.T + 2.47e6T^{2} \)
23 \( 1 + (846. + 846. i)T + 6.43e6iT^{2} \)
29 \( 1 - 6.17e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.36e3iT - 2.86e7T^{2} \)
37 \( 1 + (5.03e3 + 5.03e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.90e4T + 1.15e8T^{2} \)
43 \( 1 + (-9.23e3 - 9.23e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-1.85e4 + 1.85e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-6.80e3 + 6.80e3i)T - 4.18e8iT^{2} \)
59 \( 1 + 4.70e4T + 7.14e8T^{2} \)
61 \( 1 - 1.43e4T + 8.44e8T^{2} \)
67 \( 1 + (-3.03e4 + 3.03e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 4.39e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.98e4 - 2.98e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 5.41e4T + 3.07e9T^{2} \)
83 \( 1 + (4.50e3 + 4.50e3i)T + 3.93e9iT^{2} \)
89 \( 1 + 2.92e4iT - 5.58e9T^{2} \)
97 \( 1 + (6.39e4 + 6.39e4i)T + 8.58e9iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.32955067994985878694990169392, −11.13156553904709824197764442657, −10.28350170181769531958647751026, −9.049089754848836393313171246364, −7.38876099205091600574892459983, −6.80423797427669810397779185707, −5.70539606471990670827486198505, −4.68034084459160692865882506266, −2.18885262086706572437271330441, −1.21136735400136449434078161029, 0.49613920333697916330959823053, 2.58282409155877327523909912555, 4.42276413577471750910025679948, 5.50285464744197596016305427289, 5.91965421424798224843553211005, 7.70875982595126037852732345444, 9.300501379776662571975822551104, 9.981812004995034412136480812103, 10.74523570998477215883834830742, 11.76962004278627995769457278540

Graph of the $Z$-function along the critical line